Vertex of an Angle
GeometryThe vertex of an angle is the common endpoint shared by the two rays that form the angle.
Definition
The vertex of an angle is the corner point where the two sides (rays) of the angle meet. Every angle has exactly one vertex.
Example
In angle $ABC$, the vertex is point $B$ - it is the middle letter and the point where the two rays meet. The corner of a triangle is a vertex. A square has four vertices (corners).
Key Insight
The word "vertex" comes from Latin meaning "highest point" or "turning point." Every polygon has vertices at its corners, and every angle has a vertex where its sides meet.
Definition
The vertex of an angle is the point where the two rays forming the angle originate. In angle notation, the vertex is always the middle letter: in angle $ABC$ (written with the angle symbol before $ABC$), $B$ is the vertex. Polygons have vertices at each corner where two sides meet.
Example
Triangle $DEF$ has three angles and three vertices: $D$, $E$, and $F$. At vertex $D$, sides $DE$ and $DF$ form angle $D$. The sum of all three angles at the vertices equals $180^\circ$.
Key Insight
In a polygon, the number of vertices equals the number of sides and the number of angles. This triple equality is a fundamental property: a polygon with $n$ sides has $n$ vertices and $n$ interior angles.
Definition
The vertex of an angle is the point of concurrency of the two bounding rays. In graph theory, "vertex" generalizes to any node of a graph. In polyhedral geometry, a vertex is a point where three or more edges meet, characterized as an extreme point of the convex hull.
Example
A convex polyhedron with $V$ vertices, $E$ edges, and $F$ faces satisfies Euler's formula $V - E + F = 2$. A cube has $8$ vertices, $12$ edges, and $6$ faces: $8 - 12 + 6 = 2$.
Key Insight
The concept of vertex unifies across dimensions: 0D (a point itself), 1D (an endpoint of a segment), 2D (corner of a polygon), 3D (corner of a polyhedron). Euler's formula connecting vertices, edges, and faces is a topological invariant, true for any convex polyhedron.